Differential Equation Models by pptfiles

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									Differential Equation Models

          Section 3.5
Impulse Response of an LTI
         System
                      H(s)



H(s) is the the Laplace transform of h(t)
With s=jω, H(jω) is the Fourier transform of h(t)

Cover Laplace transform in chapter 7 and Fourier
Transform in chapter 5.

H(s) can also be understood using the differential
equation approach.
Complex Exponential
RL Circuit




      Let y(t)=i(t) and x(t)=v(t)

   

        Differential Equation & ES 220
nth order Differential Equation
• If you use more inductors/capacitors, you will get
  an nth order linear differential equation with
  constant coefficients
Solution of Differential Equations
• Find the natural response
• Find the force Response
   – Coefficient Evaluation
Determine the Natural Response
•  




           0, since we are looking for the natural response.
      Natural Response (Cont.)
•  

      Assume yc(t)=Cest
                  Nth Order System


        Assume yc(t)=Cest




(characteristic
equation)
                                (no repeated roots)
Stability ↔Root Locations
                              (unstable)
     Stable




                 (marginally stable)
            The Force Response
• Determine the form of force solution
  from x(t)
                         




Solve for the unknown coefficients Pi by substituting yp(t) into

                             
Finding The Forced Solution
Finding the General Solution



                    (initial condition)
       Nth order LTI system
• If there are more inductors and
  capacitors in the circuit,
Transfer Function




(Transfer function)
Summary (p. 125)
Summary (p. 129)

								
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